A131526 Number of degree-n permutations such that number of cycles of size 2k is even (or zero) and number of cycles of size 2k-1 is odd (or zero), for every k.
1, 1, 0, 3, 11, 40, 184, 1036, 12949, 88488, 807008, 7362586, 113572183, 1238477032, 15630890560, 228998728050, 4141605806441, 62222251093216, 1030119451142656, 19050688698470434, 412037845709792107, 8102391640556570616, 165794307361686866432
Offset: 0
Examples
a(4)=11 because we have (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3), (143)(2), (12)(34), (13)(24) and (14)(23).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
Programs
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Maple
g:=(product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..40))*(product(cosh(x^(2*k)/(2*k)), k=1..40)): gser:=series(g,x=0,25); seq(factorial(n)*coeff(gser,x,n),n=0..21); # Emeric Deutsch, Aug 28 2007 # second Maple program: with(combinat): b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add( `if`(j=0 or irem(i+j, 2)=0, multinomial(n, n-i*j, i$j)* (i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i))) end: a:= n-> b(n$2): seq(a(n), n=0..30); # Alois P. Heinz, Mar 09 2015
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Mathematica
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i + j, 2] == 0, multinomial[n, {n - i j} ~Join~ Table[i, {j}]] (i - 1)!^j/j! b[n - i j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; a /@ Range[0, 30] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)
Formula
E.g.f.: Product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..infinity) *Product(cosh(x^(2*k)/(2*k)), k=1..infinity).
Extensions
More terms from Emeric Deutsch, Aug 28 2007